How do you animate the labels for a diagram in Blender?
To create animated lines like in the video you linked, you could use Curves and animate the start and end bevel factors. (see this related question)


Add a curve object
Set the Fill to Full in Object Data > Shape


Increase the depth in Object data > Geometry > Bevel:


Animate the start and end bevel factors by right clicking and selecting insert keyframe, or pressing I while hovering over it.


E.g. by animating the Start bevel factor from 1 to 0: